Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 518343, 11 pages http://dx.doi.org/10.1155/2014/518343
Research Article Convergence of Variational Iteration Method for Solving Singular Partial Differential Equations of Fractional Order Asma Ali Elbeleze,1 Adem KJlJçman,2 and Bachok M. Taib1 1 2
Faculty of Science and Technology, Universiti Sains Islam Malaysia, 71800 Nilai, Malaysia Department of Mathematics, Faculty of Science, University Putra Malaysia, 4300 Serdang, Selangor, Malaysia
Correspondence should be addressed to Adem Kılıc¸man;
[email protected] Received 5 March 2014; Revised 16 May 2014; Accepted 10 June 2014; Published 16 July 2014 Academic Editor: Dumitru Baleanu Copyright © 2014 Asma Ali Elbeleze et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We are concerned here with singular partial differential equations of fractional order (FSPDEs). The variational iteration method (VIM) is applied to obtain approximate solutions of this type of equations. Convergence analysis of the VIM is discussed. This analysis is used to estimate the maximum absolute truncated error of the series solution. A comparison between the results of VIM solutions and exact solution is given. The fractional derivatives are described in Caputo sense.
1. Introduction In recent years, considerable attention has been devoted to the study of the fractional calculus and its numerous applications in many areas such as physics and engineering. The applications of fractional calculus used in many fields such as electrical networks, control theory of dynamical systems, probability and statistics, electrochemistry of corrosion, chemical physics, optics, and signal processing can be successfully modeled by linear or nonlinear FDEs [1–7]. Further, fractional partial differential equations appeared in many fields of engineering and science, including fractals theory, statistics, fluid flow, control theory, biology, chemistry, diffusion, probability, and potential theory [8, 9]. The singular partial differential equations of fractional order (FSPDEs), as generalizations of classical singular partial differential equations of integer order (SPDEs), are increasingly used to model problems in physics and engineering. Consequently, considerable attention has been given to the solution of singular partial differential equations of fractional order. Finding approximate or exact solutions of SPDEs is an important task. Except for a limited number of these equations, we have difficulty in finding their analytical solutions. Therefore, there have been attempts to find methods
for obtaining approximate solutions. Several such techniques have drawn special attention, such as variational iteration method [10], homotopy analysis method [11], and homotopy iteration method [12]. The variational iteration method (VIM) was proposed by He [13–16] due to its flexibility and convergence and efficiently works with different types of linear and nonlinear partial differential equations of fractional order and gives approximate analytical solution for all these types of equations without linearization or discretization; many author have been studying it; for example, see [17–21]. In this paper, we discuss the VIM for solving FSPDEs and obtain the convergence results of this method. The contribution of this work can be summarized in three points. (1) Based on the sufficient condition that guarantees the existence of a unique solution to our problem (see Theorem 6) and using the series solution, convergence of VIM is discussed (see Theorem 7). (2) Using point one, the maximum absolute truncated error of series solution of VIM is estimated (see Theorem 8). (3) Some numerical examples are given.
2
Abstract and Applied Analysis
Consider fractional singular partial differential equations with variable coefficients 𝜕𝛼 𝑢 𝜕4 𝑢 𝜕4 𝑢 𝜕4 𝑢 + 𝜇 (𝑥) 4 + 𝜆 (𝑦) 4 + ℎ (𝑧) 4 = 0, 𝛼 𝜕𝑡 𝜕𝑥 𝜕𝑦 𝜕𝑧
(1)
𝑎 < 𝑥, 𝑦, 𝑧 < 𝑏, 𝑡 > 0, where the variable coefficients subject to initial conditions
Definition 1. A real function 𝑓(𝑥), 𝑥 > 0 is said to be in space 𝐶𝜇, 𝜇 ∈ 𝑅 if there exists a real number 𝑝 > 𝜇, such that 𝑓(𝑥) = 𝑥𝑝 𝑓1 (𝑥), where 𝑓1 (𝑥) ∈ 𝐶(0, ∞), and it is said to be in the space 𝐶𝜇𝑛 if 𝑓𝑛 ∈ 𝑅𝜇 , 𝑛 ∈ 𝑁. Definition 2. The Riemann-Liouville fractional integral operator of order 𝛼 ≥ 0 of a function 𝑓 ∈ 𝐶𝜇, 𝜇 ≥ −1 is defined as 𝐽𝛼 𝑓 (𝑥) =
𝑢 (𝑥, 𝑦, 𝑧, 0) = 𝑓0 (𝑥, 𝑦, 𝑧) , 𝜕𝑢 (𝑥, 𝑦, 𝑧, 0) = 𝑓1 (𝑥, 𝑦, 𝑧) 𝜕𝑡
𝑥 1 ∫ (𝑥 − 𝑡)𝛼−1 𝑓 (𝑡) 𝑑𝑡, Γ (𝛼) 0
(2)
𝛼 > 0, 𝑡 > 0. In particular 𝐽0 𝑓(𝑥) = 𝑓(𝑥).
and boundary conditions
For 𝛽 ≥ 0 and 𝛾 ≥ −1, some properties of the operator 𝐽𝛼
𝑢 (𝑎, 𝑦, 𝑧, 𝑡) = 𝑔0 (𝑦, 𝑧, 𝑡) ,
𝑢 (𝑏, 𝑦, 𝑧, 𝑡) = 𝑔1 (𝑦, 𝑧, 𝑡)
𝑢 (𝑥, 𝑎, 𝑧, 𝑡) = 𝑔2 (𝑥, 𝑧, 𝑡) ,
𝑢 (𝑥, 𝑏, 𝑧, 𝑡) = 𝑔3 (𝑥, 𝑧, 𝑡)
(1) 𝐽𝛼 𝐽𝛽 𝑓(𝑥) = 𝐽𝛼+𝛽 𝑓(𝑥),
𝑢 (𝑥, 𝑦, 𝑎, 𝑡) = 𝑔4 (𝑥, 𝑧, 𝑡) ,
𝑢 (𝑥, 𝑦, 𝑏, 𝑡) = 𝑔5 (𝑥, 𝑧, 𝑡)
(2) 𝐽𝛼 𝐽𝛽 𝑓(𝑥) = 𝐽𝛽 𝐽𝛼 𝑓(𝑥),
𝜕2 𝑢 (𝑎, 𝑦, 𝑧, 𝑡) = 𝑘0 (𝑦, 𝑧, 𝑡) , 𝜕𝑥2
𝑥 1 ∫ (𝑥 − 𝑡)𝑚−𝛼−1 𝑓𝑚 (𝑡) 𝑑𝑡, Γ (𝑚 − 𝛼) 0
(6)
𝑚 − 1 < 𝛼 ≤ 𝑚. Lemma 4. If 𝑚 − 1 < 𝛼 ≤ 𝑚, 𝑚 ∈ 𝑁, 𝑓 ∈ 𝐶𝜇𝑚 , 𝜇 > −1, then the following two properties hold:
𝜕2 𝑢 (𝑥, 𝑎, 𝑧, 𝑡) = 𝑘3 (𝑥, 𝑧, 𝑡) , 𝜕𝑦2
(1) 𝐷𝛼 [𝐽𝛼 𝑓(𝑥)] = 𝑓(𝑥),
𝜕2 𝑢 (𝑥, 𝑦, 𝑎, 𝑡) = 𝑘4 (𝑥, 𝑦, 𝑡) , 𝜕𝑧2
𝑘 𝑘 (2) 𝐽𝛼 [𝐷𝛼 𝑓(𝑥)] = 𝑓(𝑥) − ∑𝑚−1 𝑘=1 𝑓 (0)(𝑥 /𝑘!).
(3)
where 𝜕𝛼 /𝜕𝑡𝛼 is the fractional derivative in the Caputo sense, 𝑎 < 𝑥, 𝑦, 𝑧 < 𝑏, and 𝑔𝑖 and 𝑘𝑖 , 𝑖 = 0, . . . , 5 are continuous. The 𝜕4 𝑢/𝜕𝑥4 , 𝜕4 𝑢/𝜕𝑦4 , and 𝜕4 𝑢/𝜕𝑧4 are linear bounded operator; that is, it is possible to find numbers 𝑚1 , 𝑚2 , 𝑚3 > 0 such that ‖𝜕4 𝑢/𝜕𝑥4 ‖ ≤ 𝑚1 ‖𝑢‖, ‖𝜕4 𝑢/𝜕𝑦4 ‖ ≤ 𝑚2 ‖𝑢‖, ‖𝜕4 𝑢/𝜕𝑧4 ‖ ≤ 𝑚3 ‖𝑢‖. Equation (1) can be written as 𝐷𝑡𝛼 𝑢 (𝑥, 𝑦, 𝑧, 𝑡) = 𝑓 (𝑡, 𝑢 (𝑥, 𝑦, 𝑧, 𝑡) , 𝐷𝑥𝑛1 𝑢 (𝑥, 𝑦, 𝑧, 𝑡) ,
(3) 𝐽𝛼 𝑥𝛾 = (Γ(𝛾 + 1)/Γ(𝛼 + 𝛾 + 1))𝑥𝛼+𝛾 .
𝐷𝛼 𝑓 (𝑥) =
𝜕2 𝑢 (𝑥, 𝑎, 𝑧, 𝑡) = 𝑘2 (𝑥, 𝑧, 𝑡) , 𝜕𝑦2
𝑐
are
𝑚 Definition 3. The Caputo fractional derivative of 𝑓 ∈ 𝐶−1 , 𝑚 ∈ 𝑁 is defined as
𝜕2 𝑢 (𝑏, 𝑦, 𝑧, 𝑡) = 𝑘0 (𝑦, 𝑧, 𝑡) , 𝜕𝑥2
𝜕2 𝑢 (𝑥, 𝑦, 𝑏, 𝑡) = 𝑘5 (𝑦, 𝑧, 𝑡) , 𝜕𝑧2
(5)
(4)
𝐷𝑦𝑛2 𝑢 (𝑥, 𝑦, 𝑧, 𝑡) , 𝐷𝑧𝑛3 𝑢 (𝑥, 𝑦, 𝑧, 𝑡)) , where 𝑛1 = 𝑛2 = 𝑛3 = 4.
Lemma 5. Suppose that 𝑢 and their partial derivatives are continuous; then the fractional derivative, 𝑐 𝐷𝑡𝛼 𝑢(𝑥, 𝑦, 𝑧, 𝑡), is bonded. Proof. We need to prove that it is possible to find number 𝑀 > 0 such that ‖ 𝑐 𝐷𝑡𝛼 𝑢(𝑥, 𝑦, 𝑧, 𝑡)‖ ≤ 𝑀‖𝑢‖. From the definition of Caputo fractional derivative above we have 𝑐 𝛼 𝐷𝑡 𝑢 (𝑥, 𝑦, 𝑧, 𝑡) 𝑏 1 = ∫ (𝑥 − 𝑡)𝑚−𝛼−1 𝑢(𝑚) (𝑡) 𝑑𝑡 Γ (𝑚 − 𝛼) 𝑎 ≤
(7)
|𝑏 − 𝑎| ‖𝑢‖ = 𝑀 ‖𝑢‖ , |(𝑚 − 𝛼) Γ (𝑚 − 𝛼)|
where 𝑀 = |𝑏 − 𝑎|/|(𝑚 − 𝛼)Γ(𝑚 − 𝛼)|.
2. Preliminaries
3. Analysis of the Variational Iteration Method
In this section, we give some basic definitions and properties of fractional calculus theory used in this paper.
To solve the fractional singular partial differential equations (4) by using the variational iteration method, with initial and
Abstract and Applied Analysis
3
boundary conditions (2) and (3), where ‖ 𝑐 𝐷𝑡𝛼 𝑢(𝑡)‖ = 𝑀‖𝑢‖, we construct the following correction functional:
We obtain the following iteration formula by substitution of (11) in (9) 𝑢𝑛+1 (𝑥, 𝑦, 𝑧, 𝑡)
𝑢𝑛+1 (𝑥, 𝑦, 𝑧, 𝑡) = 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑡)
= 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑡) +
𝛼
+ 𝐽𝑡𝛼 [ 𝑐 𝐷𝑡 𝑢 (𝑥, 𝑦, 𝑧, 𝑡) − 𝑓 ((𝑥, 𝑦, 𝑧, 𝑡) , 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑡) ,
1 Γ (𝛼 − 1)
𝑡
× ∫ (𝑡 − 𝑠)𝛼−2 (𝑡 − 𝑠)
(8)
(12)
0
× ( 𝑐 𝐷𝑡𝛼 𝑢 (𝑥, 𝑦, 𝑧, 𝑠)
𝐷𝑥𝑛1 𝑢 (𝑥, 𝑦, 𝑧, 𝑡) , 𝐷𝑦𝑛2 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑡) ,
−𝑓 (𝑠, 𝑢𝑘 (𝑥, 𝑦, 𝑧, 𝑠) , 𝐷𝑥𝑛1 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑠) ,
𝐷𝑧𝑛3 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑡))]
𝐷𝑦𝑛2 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑠) , 𝐷𝑧𝑛3 𝑢𝑛 (𝑡))) 𝑑𝑠.
or 𝑢𝑛+1 (𝑥, 𝑦, 𝑧, 𝑡) = 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑡) +
That is, 1 Γ (𝛼)
𝑢𝑛+1 (𝑥, 𝑦, 𝑧, 𝑡) = 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑡) −
𝑡
× ∫ (𝑡 − 𝑠)𝛼−1 𝜆 (𝑠) 0
(𝛼 − 1) Γ (𝛼)
𝑡
×(
𝑐
× ∫ (𝑡 − 𝑠)𝛼−1
𝐷𝑡𝛼 𝑢 (𝑥, 𝑦, 𝑧, 𝑠)
0
− 𝑓 (𝑠, 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑠) , 𝐷𝑥𝑛1 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑠) ,
× ( 𝑐 𝐷𝑡𝛼 𝑢 (𝑥, 𝑦, 𝑧, 𝑠)
𝐷𝑦𝑛2 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑠) , 𝐷𝑧𝑛3 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑠)) ) 𝑑𝑠. (9) 𝐽𝑡𝛼 is the Riemann-Liouville fractional integral operator of order 𝛼, with respect to variable 𝑡, and 𝜆 is a general Lagrange multiplier which can be identified as optimally variational theory [22], and 𝑢̃𝑛 (𝑥, 𝑡) are considered as restricted variation; that is, 𝛿̃ 𝑢𝑛 (𝑥, 𝑡) = 0. Making the above correction functional stationary, the following condition can be obtained: 𝛿𝑢𝑘+1 (𝑥, 𝑦, 𝑧, 𝑡)
× ∫ (𝑡 − 𝑠) 0
×(
𝑐
𝛼−1
− 𝑓 (𝑠, 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑠) , 𝐷𝑥𝑛1 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑠) , 𝐷𝑦𝑛2 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑠) , 𝐷𝑧𝑛3 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑠))) 𝑑𝑠. This yields the following iteration formula: 𝑢𝑛+1 (𝑥, 𝑦, 𝑧, 𝑡) = 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑡) − (𝛼 − 1)
1 𝛿 = 𝛿𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑡) + Γ (𝛼) 𝑡
× 𝐽𝑡𝛼 [𝑐 𝐷𝑡𝛼 𝑢 (𝑥, 𝑦, 𝑧, 𝑡) − 𝑓 (𝑡, 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑡) ,
𝜆 (𝑠)
𝐷𝑧𝑛3 𝑢𝑛 (𝑡))] .
− 𝑓 (𝑡, 𝑢̃𝑛 (𝑥, 𝑦, 𝑧, 𝑠) , 𝐷𝑥𝑛1 𝑢̃𝑛 (𝑥, 𝑦, 𝑧, 𝑠) , (𝑥, 𝑦, 𝑧, 𝑠) , 𝐷𝑧𝑛3 𝑢̃𝑛
(𝑥, 𝑦, 𝑧, 𝑠))) 𝑑𝑠 (10)
The initial approximation 𝑢0 can be chosen by the following manner which satisfies initial conditions: 1
𝑢0 = ∑ 𝛾𝑗 𝑗=0
and yields to Lagrange multiplier 𝜆 (𝑠) = 𝑠 − 𝑡.
(14)
𝐷𝑥𝑛1 𝑢 (𝑥, 𝑦, 𝑧, 𝑡) , 𝐷𝑦𝑛2 𝑢𝑛 (𝑥, 𝑦, 𝑧, 𝑡) ,
𝐷𝑡𝛼 𝑢 (𝑥, 𝑦, 𝑧, 𝑠)
𝐷𝑦𝑛2 𝑢̃𝑛
(13)
(11)
𝑡𝑗 = 𝛾0 + 𝛾1 𝑡, 𝑗!
where 𝛾0 = 𝑓0 (𝑥, 𝑦, 𝑧) , 𝛾1 = 𝑓1 (𝑥, 𝑦, 𝑧) .
(15)
4
Abstract and Applied Analysis
We can obtain the following first-order approximation by substitution of (15) into (14)
The mapping 𝐹 : 𝑋 → 𝑋 is defined as 𝑚−1
𝐹 (𝑢) = 𝑓 (𝑡, ∑ 𝑐𝑗
𝑢1 (𝑥, 𝑦, 𝑧, 𝑡)
𝑗=0
= 𝑢0 (𝑥, 𝑦, 𝑧, 𝑡) − (𝛼 − 1) 𝐽𝑡𝛼
(20) Let 𝑢, V ∈ 𝑋; then
× [𝑐 𝐷𝑡𝛼 𝑢0 (𝑥, 𝑦, 𝑧, 𝑡) (16)
− 𝑓 (𝑡, 𝑢0 (𝑡) ,
|𝐹 (𝑢) − 𝐹 (V)| 𝑚−1 𝑡𝑗 = 𝑓 (𝑡, ∑ 𝑐𝑗 + 𝐽𝛼 𝑢, 𝐽𝛼 𝐷𝑥𝑛1 𝑢, 𝐽𝛼 𝐷𝑦𝑛2 𝑢, 𝐽𝛼 𝐷𝑧𝑛3 𝑢) 𝑗=0 𝑗!
𝐷𝑥𝑛1 𝑢 (𝑥, 𝑦, 𝑧, 𝑡) , 𝐷𝑦𝑛2 𝑢0 (𝑥, 𝑦, 𝑧, 𝑡) , 𝐷𝑧3𝑚 𝑢0 (𝑥, 𝑦, 𝑧, 𝑡))] .
𝑡𝑗 𝛼 𝛼 𝑛1 𝛼 𝑛2 𝛼 𝑛3 −𝑓 (𝑡, ∑ 𝑐𝑗 + 𝐽 V, 𝐽 𝐷𝑥 V, 𝐽 𝐷𝑦 V, 𝐽 𝐷𝑧 V) 𝑗=0 𝑗! 𝑚−1
Finally, by substituting the constant values of 𝛾0 and 𝛾1 into (16), we have the results as the first approximate solutions of (4) with (2) and (3).
3
≤ 𝐿∑ 𝐽𝛼−𝑛𝑖 𝑢 − 𝐽𝛼−𝑛𝑖 V 𝑖=0
3.1. Convergence Analysis 3.1.1. Existence and Uniqueness Theorem. Define 𝐹 : 𝑋 → 𝑋 contentious mapping, and the function 𝐹(𝑡, 𝑢0 , 𝑢1 , . . . , 𝑢𝑛−1 ) exists with continuous and bounded derivatives, where 𝑋 is the Banach space (𝐶(𝐽), ‖ ⋅ ‖), the space of all continuous functions on 𝐽 with the norm
3 𝑡 1 ≤ 𝐿∑ ∫ (𝑡 − 𝑠)𝛼−𝑛𝑖 −1 [𝑢 − V] 𝑑𝑠 𝑖=0 Γ (𝛼 − 𝑛𝑖 ) 0 max |𝐹 (𝑢) − 𝐹 (V)| 𝑡 1 max ∫ (𝑡 − 𝑠)𝛼−𝑛𝑖 −1 [𝑢 − V] 𝑑𝑠 Γ (𝛼 − 𝑛 ) 0 𝑖 𝑖=0 3
(17)
≤ 𝐿∑
and satisfies Lipschitz condition with Lipschitz constant 𝐿, such that 𝑛 𝑓 (𝑡, 𝑢1 (𝑥, 𝑦, 𝑧, 𝑡) , 𝐷 1 𝑢1 (𝑥, 𝑦, 𝑧, 𝑡) ,
≤ 𝐿∑
‖𝑢‖ = max |𝑢| , ∀𝑡∈𝐽
3
LMT ‖𝑢 − V‖ Γ (𝛼 − 𝑛𝑖 ) 𝑖=0
− 𝑓 (𝑡, 𝑢2 (𝑥, 𝑦, 𝑧, 𝑡) , 𝐷𝑛1 𝑢2 (𝑥, 𝑦, 𝑧, 𝑡) ,
≤ 𝛾 ‖𝑢 − V‖ ,
𝐷 𝑢2 (𝑥, 𝑦, 𝑧, 𝑡) , 𝐷 𝑢2 (𝑥, 𝑦, 𝑧, 𝑡)) 𝑛𝑚
≤ 𝐿 (𝑢1 (𝑥, 𝑦, 𝑧, 𝑡) , 𝐷𝑛1 𝑢1 (𝑥, 𝑦, 𝑧, 𝑡) ,
‖𝑢 − V‖ 𝑡 𝛼−𝑛 −1 ∫ (𝑡 − 𝑠) 𝑖 𝑑𝑠 𝑖=0 Γ (𝛼 − 𝑛𝑖 ) 0 3
≤∑
𝐷𝑛2 𝑢1 (𝑥, 𝑦, 𝑧, 𝑡) , 𝐷𝑛3 𝑢1 (𝑥, 𝑦, 𝑧, 𝑡))
𝑛2
𝑡𝑗 + 𝐽𝛼 𝑢, 𝐽𝛼 𝐷𝑥𝑛1 𝑢, 𝐽𝛼 𝐷𝑦𝑛2 𝑢, 𝐽𝛼 𝐷𝑧𝑛3 𝑢) . 𝑗!
(21) (18)
𝐷𝑛2 𝑢1 𝑥, 𝑦, 𝑧, (𝑡) , 𝐷𝑛3 𝑢1 (𝑥, 𝑦, 𝑧, 𝑡)) − (𝑢2 (𝑡) , 𝐷𝑛1 𝑢2 (𝑥, 𝑦, 𝑧, 𝑡) , 𝐷𝑛2 𝑢2 (𝑥, 𝑦, 𝑧, 𝑡) , 𝐷𝑛3 𝑢2 (𝑥, 𝑦, 𝑧, 𝑡))
where 𝛾 = ∑3𝑖=0 (LMT/(Γ(𝛼 − 𝑛𝑖 ))) < 1, then we get ‖𝐹 (𝑢) − 𝐹 (V)‖ ≤ ‖𝑢 − V‖ ,
(22)
therefore the mapping 𝐹 is contraction, and there exists unique solution 𝑢 ∈ 𝐶(𝐽) to problem (4). (2) The uniqueness of the solution (see [23]).
0 < 𝐿 < 1, 𝑡 ≥ 0. Theorem 6. Let 𝑓 satisfy the Lipschitz condition (18) then the problem (4) with (2) and (3) has unique solution 𝑢(𝑥, 𝑡), whenever 0 < 𝐿 < 1. Proof. (1) The existence of the solution. From equation (4) we have 𝑚−1
𝑢 = 𝑓 (𝑡, ∑ 𝑐𝑗 𝑗=0
𝑡𝑗 + 𝐽𝛼 𝑢, 𝐽𝛼 𝐷𝑥𝑛1 𝑢, 𝐽𝛼 𝐷𝑦𝑛2 𝑢, 𝐽𝛼 𝐷𝑧𝑛3 𝑢) . (19) 𝑗!
3.1.2. Proof of Convergence Theorem 7. Suppose that 𝑋 is Banach space and 𝐹 : 𝑋 → 𝑋 satisfies condition (18). Then, the sequence (14) converges to the solution of (4) with (2) and (3). Proof. Defined (𝐶(𝐽), ‖ ⋅ ‖) is the Banach space, the space of all continuous functions on 𝐽 with the norm 𝑢 (𝑥, 𝑦, 𝑧, 𝑡) = max 𝑢 (𝑥, 𝑦, 𝑧, 𝑡) . (23) ∀𝑡∈𝐽
Abstract and Applied Analysis
5
We need to show that {𝑢𝑛 } is a Cauchy sequence in this Banach space: 𝑢𝑛 − 𝑢𝑚 = max 𝑢𝑛 − 𝑢𝑚 (𝛼 − 1) = max 𝑢𝑛−1 − Γ (𝛼)
𝑅 = max (𝑡 − 𝑠)𝛼−1 . 0≤𝑠≤𝑡,0≤𝑡≤𝑇
(25)
Finally, we have 𝑢𝑛 − 𝑢𝑚 ≤ (1 −
𝑡
× ∫ (𝑡 − 𝑠)𝛼−1
(𝑀 + (𝑚1 + 𝑚2 + 𝑚3 ) 𝑅𝑇) ) Γ (𝛼 − 1)
× 𝑢𝑛−1 − 𝑢𝑚−1 ≤ 𝛾 𝑢𝑛−1 − 𝑢𝑚−1 ,
0
× [𝐷𝛼 𝑢𝑛−1 (𝑥, 𝑦, 𝑧, 𝑠)
𝛾 = (1 −
𝐷𝑦𝑛2 𝑢𝑛−1 (𝑠) , 𝐷𝑧𝑛3 𝑢𝑛−1 (𝑠))] 𝑑𝑠 (𝛼 − 1) Γ (𝛼)
(𝑀 + (𝑚1 + 𝑚2 + 𝑚3 ) 𝑅𝑇) ). Γ (𝛼 − 1)
(27)
Let 𝑛 = 𝑚 + 1. Then 𝑢𝑚+1 − 𝑢𝑚
𝑡
≤ 𝛾 𝑢𝑚 − 𝑢𝑚−1 ≤ 𝛾2 𝑢𝑚−1 − 𝑢𝑚−2
× ∫ (𝑡 − 𝑠)𝛼−1 0
(28)
≤ ⋅ ⋅ ⋅ ≤ 𝛾𝑚 𝑢1 − 𝑢0 .
× [𝐷𝛼 𝑢𝑚−1
From the triangle inequality, we have 𝑢𝑛 − 𝑢𝑚
− 𝐹 (𝑠, 𝑢𝑚−1 (𝑥, 𝑦, 𝑧, 𝑠) 𝐷𝑥𝑛1 𝑢𝑚−1 , 𝐷𝑦𝑛2 𝑢𝑚−1 , 𝐷𝑧𝑛3 𝑢𝑚−1 )] 𝑑𝑠
≤ 𝑢𝑚+1 − 𝑢𝑚 + 𝑢𝑚+2 − 𝑢𝑚+1
≤ max [𝑢𝑛−1 − 𝑢𝑚−1
≤ ⋅ ⋅ ⋅ ≤ 𝑢𝑛 − 𝑢𝑛−1 ≤ 𝛾𝑚 𝑢1 − 𝑢0 + 𝛾𝑚+1 𝑢1 − 𝑢0
(𝛼 − 1) − Γ (𝛼)
+ ⋅ ⋅ ⋅ + 𝛾𝑛−1 𝑢1 − 𝑢0
𝑡
× ∫ (𝑡 − 𝑠)𝛼−1 [𝐷𝛼 𝑢𝑛−1 − 𝐷𝛼 𝑢𝑚−1 ]
(29)
≤ [𝛾𝑚 + 𝛾𝑚+1 + 𝛾𝑚+2 + ⋅ ⋅ ⋅ + 𝛾𝑛−1 ] 𝑢1 − 𝑢0
0
− 𝐹 (𝑠, 𝑢𝑛−1 , 𝐷𝑥𝑛1 𝑢𝑛−1 , 𝐷𝑦𝑛2 𝑢𝑛−1 , 𝐷𝑧𝑛3 𝑢𝑛−1 ) 𝑑𝑠 − 𝐹 (𝑠, 𝑢𝑚−1 , 𝐷𝑥𝑛1 𝑢𝑚−1 , 𝐷𝑦𝑛2 𝑢𝑚−1 , 𝐷𝑧𝑛3 𝑢𝑛−1 ) 𝑑𝑠] ≤ max [ 𝑢𝑛−1 − 𝑢𝑚−1
≤ 𝛾𝑚 [1 + 𝛾 + 𝛾2 + ⋅ ⋅ ⋅ + 𝛾𝑛−𝑚−1 ] 𝑢1 − 𝑢0 ≤ 𝛾𝑚 (
1 − 𝛾𝑛−𝑚 ) 𝑢1 − 𝑢0 . 1−𝛾
Since 0 < 𝛾 < 1, so 1 − 𝛾𝑛−𝑚 < 1, and then 𝛾𝑚 𝑢𝑛 − 𝑢𝑚 ≤ 𝑢 − 𝑢0 . 1−𝛾 1
(𝑀 + (𝑚1 + 𝑚2 + 𝑚3 ) 𝑅𝑇) − Γ (𝛼)
(30)
But ‖𝑢1 − 𝑢0 ‖ < ∞; then ‖𝑢𝑛 − 𝑢𝑚 ‖ → 0 as 𝑚 → ∞. We conclude that 𝑢𝑛 is a Cauchy sequence in 𝐶[𝐽], so the sequence converges and the proof is complete.
𝑡 × ∫ (𝑡 − 𝑠)𝛼−1 𝑢𝑛−1 − 𝑢𝑚−1 𝑑𝑠] 0
≤ max 𝑢𝑛−1 − 𝑢𝑚−1
3.1.3. Error Analysis
(𝑀 + (𝑚1 + 𝑚2 + 𝑚3 ) 𝑅𝑇) × (1 − Γ (𝛼 − 1) 𝑡 × ∫ (𝑡 − 𝑠)𝛼−1 𝑑𝑠) , 0
(26)
where 𝑀, 𝑅, 𝑇, Γ(𝛼) are constants and
− 𝐹 (𝑠, 𝑢𝑛−1 (𝑠) , 𝐷𝑥𝑛1 𝑢𝑛−1 (𝑠) ,
− 𝑢𝑚−1 +
where
(24)
Theorem 8. The maximum absolute error of the approximate solution 𝑢𝑚 to problem (4)-(3) is estimated to be max 𝑢𝑒𝑥𝑎𝑐𝑡 − 𝑢𝑛 ≤ 𝑘, (31) 𝑡∈𝐽
6
Abstract and Applied Analysis the exact solution in special case 𝛼 = 2 is
where 𝑘= (
𝑚
𝛾 (𝑀 + (𝑚1 + 𝑚2 + 𝑚3 RT) 𝛽) ) 𝑢0 , (1 − 𝛾)
𝛼 − 1 𝛽 = ( ) . Γ (𝛼)
(32)
(33)
= 𝑢𝑛 (𝑥, 𝑡) − (𝛼 − 1) 𝐽𝑡𝛼 𝜕𝛼 𝑢𝑛 (𝑥, 𝑡) 1 𝑥4 𝜕 4 𝑢 ×( + ( ). + ) 𝜕𝑡𝛼 𝑥 120 𝜕𝑥4
(𝛼 − 1) 𝑡 = max − ∫ (𝑡 − 𝑠)𝛼−1 𝑡∈𝐽 Γ (𝛼) 0 × [𝐷𝛼 𝑢0
(34)
− 𝐹 (𝑠, 𝑢0 , 𝐷𝑥𝑛1 𝑢0 , 𝐷𝑦𝑛2 , 𝐷𝑧𝑛3 𝑢0 )] 𝑑𝑠 = [(𝑀 + (𝑚1 + 𝑚2 + 𝑚3 𝑅𝑇) 𝛽)] 𝑢0 ,
(40)
Suppose that an initial approximation has the following form which satisfies the initial conditions: 𝑢0 (𝑥, 𝑡) = (1 +
where 𝛽 = |((𝛼 − 1)/Γ(𝛼))|, and thus, the maximum absolute error in the interval 𝐽 is 𝑢exact − 𝑢𝑛 ≤ max 𝑢exact − 𝑢𝑛 ≤ 𝑘. (35) 𝑡∈𝐽
𝑥5 ) 𝑡. 120
(41)
Now by iteration formula (16), we obtain the following approximations: 𝑢1 (𝑥, 𝑡) = 𝑢0 (𝑥, 𝑡) − (𝛼 − 1) 𝐽𝑡𝛼
This completes the proof.
4. Numerical Examples
×(
Example 1. Consider the following fourth-order fractional singular partial differential equation: 𝛼
4
𝑡 > 0,
𝑢 (𝑥, 0) = 0,
(36)
1 < 𝛼 ≤ 2.
1 < 𝑥 < 1, 2
𝑥5 𝜕𝑢 , (𝑥, 0) = 1 + 𝜕𝑡 120 and boundary conditions
𝜕2 𝑢 1 (1, 𝑡) = sin 𝑡, 2 𝜕𝑥 6
(37)
The second approximation takes the following form:
×(
𝜕𝛼 𝑢1 (𝑥, 𝑡) 1 𝑥4 𝜕4 𝑢1 +( + ) ) 𝛼 𝜕𝑡 𝑥 120 𝜕𝑥4
= (1 +
𝑥5 𝑥5 𝑡𝛼+1 ) 𝑡 − (𝛼 − 1) (1 + ) 120 120 Γ (𝛼 + 2)
+ (𝛼 − 1)2 (1 +
𝑡 > 0,
𝑡 > 0,
𝑥5 𝑥5 𝑡𝛼+1 . )) 𝑡 − (𝛼 − 1) (1 + ) 120 120 Γ (𝛼 + 2)
= 𝑢1 (𝑥, 𝑡) − (𝛼 − 1) 𝐽𝑡𝛼
0
